Newton’s Laws - continued
Friction, Inclined Planes, N.T.L.
TWO types of Friction


Static – Friction that keeps an object at rest
and prevents it from moving
Kinetic – Friction that acts during motion
Force of Friction

The Force of Friction is F f  F N
  constant of proportion ality
directly related to the
Force Normal.
  coefficien t of friction

Mostly due to the fact
that BOTH are surface
forces
F sf   s F N
F kf   k F N
The coefficient of
friction is a unitless
constant that is
specific to the
material type and
usually less than
one.
Note: Friction ONLY depends on the MATERIALS sliding against
each other, NOT on surface area.
Example
A 1500 N crate is being pushed
across a level floor at a
constant speed by a force F of
600 N at an angle of 20° below
the horizontal as shown in the
figure.
Fa
a) What is the coefficient of kinetic
friction between the crate and the
floor?
F f   k FN
FN
Fay
F f  F ax  F a cos   600 (cos 20 )  563 . 82 N
F N  F ay  mg  F a sin   1500
20
F N  600 (sin 20 )  1500  1705 . 21 N
Fax
563 . 82   k 1705 . 21
 k  0 . 331
Ff
mg
Example
FN
If the 600 N force is instead pulling the
block at an angle of 20° above the
horizontal as shown in the figure,
what will be the acceleration of the
crate. Assume that the coefficient of
friction is the same as found in (a)
F Net  ma
20
Fax
Ff
mg
F ax  F f  ma
F a cos    F N  ma
F a cos    ( mg  F a sin  )  ma
600 cos 20  0 . 331 (1500  600 sin 20 )  153 . 1a
563 . 8  428 . 57  153 . 1a
a  0 . 883 m / s
2
Fa
Fay
Inclines

Ff
FN
mg cos 


mg 
mg sin 


Tips
•Rotate Axis
•Break weight into components
•Write equations of motion or
equilibrium
•Solve
Example
Masses m1 = 4.00 kg and m2 = 9.00 kg are connected by a light string that passes over
a frictionless pulley. As shown in the diagram, m1 is held at rest on the floor and m2 rests
on a fixed incline of angle 40 degrees. The masses are released from rest, and m2 slides
1.00 m down the incline in 4 seconds. Determine (a) The acceleration of each mass (b)
The coefficient of kinetic friction and (c) the tension in the string.
T
FN
F NET  ma
T  m1 g  m1a  T  m1a  m1 g
Ff
m2gcos40
m 2 g sin   ( F f  T )  m 2 a
40
T
m2g
m1
40
m2gsin40
m1g
Example
F NET  ma
T  m1 g  m1a  T  m1a  m1 g
m 2 g sin   ( F f  T )  m 2 a
2
x  v ox t  1 at
2
T  4 (. 125 )  4 ( 9 . 8 )  39 . 7 N
2
1  0  1 a (4)
2
a  0 . 125 m / s
2
m 2 g sin   F f  T  m 2 a
m 2 g sin   F f  ( m 1 a  m 1 g )  m 2 a
m 2 g sin    k F N  m 1 a  m 1 g  m 2 a
m 2 g sin    k m 2 g cos   m 1 a  m 1 g  m 2 a
m 2 g sin   m 1 a  m 1 g  m 2 a   k m 2 g cos 
k 
m 2 g sin   m 1 a  m 1 g  m 2 a
m 2 g cos 
k 
56 . 7  0 . 5  39 . 2  1 . 125
67 . 57
 0 . 235
Newton’s Third Law
“For every action there is an EQUAL and
OPPOSITE reaction.


This law focuses on action/reaction pairs (forces)
They NEVER cancel out
All you do is SWITCH the wording!
•PERSON on WALL
•WALL on PERSON
N.T.L
This figure shows the force during a
collision between a truck and a train. You
can clearly see the forces are EQUAL
and OPPOSITE. To help you understand
the law better, look at this situation from
the point of view of Newton’s Second
Law.
FTruck  FTrain
m Truck ATruck  M Train a Train
There is a balance between the mass and acceleration. One object usually
has a LARGE MASS and a SMALL ACCELERATION, while the other has a
SMALL MASS (comparatively) and a LARGE ACCELERATION.
N.T.L Examples
Action: HAMMER HITS NAIL
Reaction: NAIL HITS HAMMER
Action: Earth pulls on YOU
Reaction: YOU pull on the earth
An interesting friction/calc
problem…YUCK!
Suppose you had a 30- kg box that
is moving at a constant speed
until it hits a patch of sticky
snow where it experiences a
frictional force of 12N.
a)
b)
What is the acceleration of
the box?
What is the coefficient of
kinetic friction between the
box and the snow?
F f   k F N   k mg
12   k ( 30 )( 9 . 8 )
k 
0.04
FN
Ff
mg
F Net  ma
F f  ma  12  30 a
a
0.4 m/s/s
The “not so much fun” begins….
Now suppose your friend decides to help by pulling the box
across the snow using a rope that is at some angle from the
horizontal. She begins by experimenting with the angle of pull
and decides that 40 degrees is NOT optimal. At what angle, ,
will the minimum force be required to pull the sled with a
constant velocity?
Let’s start by making a function for “F” in terms of “theta” using
our equations of motion. F  F sin   mg  F  mg  F sin 
N
FN
F cos   F F   k F N
F
F cos    K ( mg  F sin  )

Fsin F cos    mg   F sin 
k
k
Fcos
Ff
N
F cos    k F sin    k mg
F (cos    k sin  )   k mg
mg
F ( ) 
 k mg
cos    k sin 
What does this graph look like?
F ( ) 
 k mg
Theta
Force
1
11.7536
cos    k sin 
10
11.8579
20
12.3351
30
13.2728
40
14.8531
50
17.4629
60
21.9961
70
30.9793
Force vs. Theta
35
30
Force (N)
25
Does this graph tell
you the angle needed
for a minimum force?
20
15
10
5
0
0
10
20
30
40
50
Theta (degrees)
60
70
80
What does this graph look like?
Force
0.5
11.7563
1
11.7536
1.5
11.7517
2
11.7508
2.5
11.7507
3
11.7515
3.5
11.7532
4
11.7558
4.5
11.7593
5
11.7638
Could this graph tell
you the angle needed
for a minimum force?
What do you notice about
the SLOPE at this minimum
force?
 k mg
F ( ) 
cos    k sin 
Force vs. Theta
11.766
11.764
11.762
Force (N)
theta
11.76
11.758
11.756
11.754
11.752
11.75
0
1
2
3
Theta (degrees)
4
5
6
Taking the derivative
Force vs. Theta
11.766
11.764
11.762
Force (N)
Here is the point. If we find the
derivative of the function
and set the derivative equal
to ZERO, we can find the
ANGLE at this minimum.
Remember that the
derivative is the SLOPE of
the tangent line. The
tangent line’s slope is zero
at the minimum force and
thus can be used to
determine the angle we
need.
11.76
11.758
11.756
11.754
11.752
11.75
0
1
2
3
4
5
6
Theta (degrees)
This tells us that our minimum force is somewhere between 2 & 3 degrees.
Taking the derivative using the Chain Rule
F ( ) 
 k mg
cos    k sin 
d(
dF
d

 k mg
cos    k sin 
)
1

d
d (  k mg (cos    k sin  ) )
d
f ( x )  sin( 3 x  x )
2
f  ( x )  cos
(3 x  x )
2
Derivative of
Leave inside
outside function function alone
( 6 x  1)
Derivative of
inside function
2
f  ( x )  ( 6 x  1) cos( 3 x  x )
Taking the derivative using the Chain Rule
dF
d
1

d (  k mg (cos    k sin  ) )
d
 k mg  constants
- 1( cos    k sin  )
as well as leaving
2
 derivative
the inside function
 sin    k cos   derivative
F  ( ) 
of outside function
alone
of inside function
  k mg (  sin    k cos  )
( cos    k sin  )
2
Now we set the derivative equal to ZERO and solve for theta!
Setting the derivative equal to zero
F  ( ) 
0
  k mg (  sin    k cos  )
( cos    k sin  )
  k mg (  sin    k cos  )
( cos    k sin  )
2
0   sin    k cos 
sin    k cos 
tan    k
  tan
 
2
1
(  k )  tan
2.29°
1
( 0 . 04 )